This memoir develops, discusses and compares a range of com-
mutative and non-commutative invariants defined for projection
method tilings and point patterns. The projection method refers to
patterns, particularly the quasiperiodic patterns, constructed by the
projection of a strip of a high dimensional integer lattice to a smal-
ler dimensional Euclidean space. In the first half of the memoir the
acceptance domain is very general - any compact set which is the
closure of its interior - while in the second half we concentrate on the
so-called canonical patterns. The topological invariants used are vari-
ous forms of K-theory and cohomology applied to a variety of both
C*-algebras and dynamical systems derived from such a pattern.
The invariants considered all aim to capture geometric properties
of the original patterns, such as quasiperiodicity or self-similarity, but
one of the main motivations is also to provide an accessible approach
to the the KQ group of the algebra of observables associated to a
quasicrystal with atoms arranged on such a pattern.
The main results provide complete descriptions of the (un-
ordered) X-theory and cohomology of codimension 1 projection pat-
terns, formulae for these invariants for codimension 2 and 3 canonical
projection patterns, general methods for higher codimension patterns
and a closed formula for the Euler characteristic of arbitrary canon-
ical projection patterns. Computations are made for the Ammann-
Kramer tiling. Also included are qualitative descriptions of these
invariants for generic canonical projection patterns. Further results
include an obstruction to a tiling arising as a substitution and an
obstruction to a substitution pattern arising as a projection. One co-
rollary is that, generically, projection patterns cannot be derived via
substitution systems.
Mathematics Subject Classification Primary: 52C23
Mathematical Subject Classification Secondary: 19E20, 37Bxx,
46Lxx, 55Txx, 82D25
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